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A review on δ-structurable algebras

Noriaki Kamiya, Daniel Mondoc, Susumu Okubo (2011)

Banach Center Publications

In this paper we give a review on δ-structurable algebras. A connection between Malcev algebras and a generalization of δ-structurable algebras is also given.

Category 𝒪 for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

In this paper we study the BGG-categories 𝒪 q associated to quantum groups. We prove that many properties of the ordinary BGG-category 𝒪 for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when q is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for 𝒪 and for finite dimensional U q -modules we are able to determine...

Commutator algebras arising from splicing operations

Sergei Sverchkov (2014)

Open Mathematics

We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and...

Directed pseudo-graphs and Lie algebras over finite fields

Luis B. Boza, Eugenio Manuel Fedriani, Juan Núñez, Ana María Pacheco, María Trinidad Villar (2014)

Czechoslovak Mathematical Journal

The main goal of this paper is to show an application of Graph Theory to classifying Lie algebras over finite fields. It is rooted in the representation of each Lie algebra by a certain pseudo-graph. As partial results, it is deduced that there exist, up to isomorphism, four, six, fourteen and thirty-four 2 -, 3 -, 4 -, and 5 -dimensional algebras of the studied family, respectively, over the field / 2 . Over / 3 , eight and twenty-two 2 - and 3 -dimensional Lie algebras, respectively, are also found. Finally,...

Free associative algebras, noncommutative Gröbner bases, and universal associative envelopes for nonassociative structures

Murray R. Bremner (2014)

Commentationes Mathematicae Universitatis Carolinae

First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.

Hom-Akivis algebras

A. Nourou Issa (2011)

Commentationes Mathematicae Universitatis Carolinae

Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms and that the class of Hom-Akivis algebras is closed under self-morphisms. It is pointed out that a Hom-Akivis algebra associated to a Hom-alternative algebra is a Hom-Malcev algebra.

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